A function which is continuous on each path component is continuous

Question

Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a function which is continuous on each path-component of $X$. Then is $f$ necessarily continuous?

The result if clear when $X$ is locally path-connected, so that each path-component is open in $X$. But in the general case, I can't see whether this is true or not. How about the connected components?

Answer 1

No, consider the deleted comb space for a trivial example of how it can go awry.